Calculate the sphericity event shape. More...

#include <Sphericity.hh>

Inheritance diagram for Rivet::Sphericity:

Public Member Functions
void	clear ()
	Reset the projection.

Vector3	mkEigenVector (Matrix3 A, const double &lambda)

Constructors etc.
	Sphericity (double rparam=2.0)
	Constructor.

	Sphericity (const FinalState &fsp, double rparam=2.0)

	DEFAULT_RIVET_PROJ_CLONE (Sphericity)
	Clone on the heap.

Access the event shapes by name
double	sphericity () const
	Sphericity.

double	transSphericity () const
	Transverse sphericity.

double	planarity () const
	Planarity.

double	aplanarity () const
	Aplanarity.

Access the sphericity basis vectors
const Vector3 &	sphericityAxis () const
	Sphericity axis.

const Vector3 &	sphericityMajorAxis () const
	Sphericity major axis.

const Vector3 &	sphericityMinorAxis () const
	Sphericity minor axis.

AxesDefinition axis accessors
const Vector3 &	axis1 () const

const Vector3 &	axis2 () const
	The 2nd most significant ("major") axis.

const Vector3 &	axis3 () const
	The least significant ("minor") axis.

Access the momentum tensor eigenvalues
double	lambda1 () const

double	lambda2 () const

double	lambda3 () const

Direct methods
Ways to do the calculation directly, without engaging the caching system
void	calc (const FinalState &fs)
	Manually calculate the sphericity, without engaging the caching system.

void	calc (const Particles &particles)
	Manually calculate the sphericity, without engaging the caching system.

void	calc (const Jets &jets)
	Manually calculate the sphericity, without engaging the caching system.

void	calc (const vector< FourMomentum > &momenta)
	Manually calculate the sphericity, without engaging the caching system.

void	calc (const vector< Vector3 > &momenta)
	Manually calculate the sphericity, without engaging the caching system. More...

Public Member Functions inherited from Rivet::AxesDefinition
virtual	~AxesDefinition ()
	Virtual destructor.

virtual unique_ptr< Projection >	clone () const =0
	Clone on the heap.

Public Member Functions inherited from Rivet::Projection
virtual std::string	name () const
	Get the name of the projection.

	Projection ()
	The default constructor.

virtual	~Projection ()
	The destructor.

bool	before (const Projection &p) const

virtual const std::set< PdgIdPair >	beamPairs () const

Projection &	addPdgIdPair (PdgId beam1, PdgId beam2)

Public Member Functions inherited from Rivet::ProjectionApplier
	ProjectionApplier ()
	Constructor.

void	markAsOwned () const
	Mark this object as owned by a proj-handler.

std::set< ConstProjectionPtr >	getProjections () const
	Get the contained projections, including recursion.

bool	hasProjection (const std::string &name) const
	Does this applier have a projection registered under the name name?

template<typename PROJ >
const PROJ &	getProjection (const std::string &name) const

template<typename PROJ >
const PROJ &	get (const std::string &name) const

const Projection &	getProjection (const std::string &name) const

template<typename PROJ >
const PROJ &	applyProjection (const Event &evt, const Projection &proj) const
	Apply the supplied projection on event evt. More...

template<typename PROJ >
const PROJ &	apply (const Event &evt, const Projection &proj) const

template<typename PROJ >
const PROJ &	applyProjection (const Event &evt, const PROJ &proj) const
	Apply the supplied projection on event evt. More...

template<typename PROJ >
const PROJ &	apply (const Event &evt, const PROJ &proj) const

template<typename PROJ >
const PROJ &	applyProjection (const Event &evt, const std::string &name) const

template<typename PROJ >
const PROJ &	apply (const Event &evt, const std::string &name) const

template<typename PROJ >
const PROJ &	apply (const std::string &name, const Event &evt) const

Protected Member Functions
void	project (const Event &e)
	Perform the projection on the Event.

int	compare (const Projection &p) const
	Compare with other projections.

Protected Member Functions inherited from Rivet::Projection
Log &	getLog () const
	Get a Log object based on the getName() property of the calling projection object.

void	setName (const std::string &name)
	Used by derived classes to set their name.

Cmp< Projection >	mkNamedPCmp (const Projection &otherparent, const std::string &pname) const

Cmp< Projection >	mkPCmp (const Projection &otherparent, const std::string &pname) const

virtual Projection &	operator= (const Projection &)
	Block Projection copying.

Protected Member Functions inherited from Rivet::ProjectionApplier
Log &	getLog () const

ProjectionHandler &	getProjHandler () const
	Get a reference to the ProjectionHandler for this thread.

template<typename PROJ >
const PROJ &	declareProjection (const PROJ &proj, const std::string &name)
	Register a contained projection. More...

template<typename PROJ >
const PROJ &	declare (const PROJ &proj, const std::string &name)
	Register a contained projection (user-facing version) More...

template<typename PROJ >
const PROJ &	declare (const std::string &name, const PROJ &proj)
	Register a contained projection (user-facing, arg-reordered version) More...

template<typename PROJ >
const PROJ &	addProjection (const PROJ &proj, const std::string &name)
	Register a contained projection (user-facing version) More...

Detailed Description

Calculate the sphericity event shape.

The sphericity tensor (or quadratic momentum tensor) is defined as

$S^{\alpha \beta} = \frac{\sum_i p_i^\alpha p_i^\beta}{\sum_i |\mathbf{p}_i|^2}$

, where the Greek indices are spatial components and the Latin indices are used for sums over particles. From this, the sphericity, aplanarity and planarity can be calculated by combinations of eigenvalues.

Defining the three eigenvalues $\lambda_1 \ge \lambda_2 \ge \lambda_3$ , with $\lambda_1 + \lambda_2 + \lambda_3 = 1$ , the sphericity is

$S = \frac{3}{2} (\lambda_2 + \lambda_3)$

The aplanarity is $A = \frac{3}{2}\lambda_3$ and the planarity is $P = \frac{2}{3}(S-2A) = \lambda_2 - \lambda_3$ . The eigenvectors define a set of spatial axes comparable with the thrust axes, but more sensitive to high momentum particles due to the quadratic sensitivity of the tensor to the particle momenta.

Since the sphericity is quadratic in the particle momenta, it is not an infrared safe observable in perturbative QCD. This can be fixed by adding a regularizing power of $r$ to the definition:

$S^{\alpha \beta} = \frac{\sum_i |\mathbf{p}_i|^{r-2} p_i^\alpha p_i^\beta} {\sum_i |\mathbf{p}_i|^r}$

$r$ is available as a constructor argument on this class and will be taken into account by the Cmp<Projection> operation, so a single analysis can use several sphericity projections with different $r$ values without fear of a clash.